The Ising model, named after the physicist Ernst Ising, is a mathematical model in statistical mechanics. It can be represented on a graph where its configuration space is the set of all possible assignments of +1 or -1 to each vertex of the graph. To complete the model, a function, E(e) must be defined, giving the difference between the energy of the "bond" associated with the edge when the spins on both ends of the bond are opposite and the energy when they are aligned. It's also possible to have an external magnetic field. Ernst Ising (born May 10, 1900, Cologne Germany – May 11, 1998, Peoria, Illinois) was a German physicist, who is best remembered for the development of the Ising model of ferromagnetism. ... Mathematical models are of great importance in physics. ... Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in most applications, it is a topological space or/and a vector space. ... A diagram of a graph with 6 vertices and 7 edges. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... Edge may have one of the following special meanings, in addition to its dictionary definition: wiktionary:edge. ... In physics, spin is an intrinsic angular momentum associated with microscopic particles. ... Current flowing through a wire produces a magnetic field (M) around the wire. ...

At a finite temperature, T, the probability of a configuration is proportional to Temperature is the physical property of a system which underlies the common notions of hot and cold; the material with the higher temperature is said to be hotter. ... The word probability derives from the Latin probare (to prove, or to test). ... The word proportionality may have one of a number of meanings: In mathematics, proportionality is a mathematical relation between two quantities. ...

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See partition function (statistical mechanics). A full mathematical development of the Ising model and its solution in 1D is given in the article on the Potts model. In statistical mechanics, the partition function Z is an important quantity that encodes the statistical properties of a system in thermodynamic equilibrium. ... In statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. ...

In his 1925 PhD thesis, Ising solved the model for the 1D case. In one dimension, the solution admits no phase transition. On the basis of this result, he incorrectly concluded that his model does not exhibit phase behavior in any dimension. 1925 (MCMXXV) was a common year starting on Thursday (link will take you to calendar). ... In physics, a phase transition is the transformation of a thermodynamic system from one phase to another. ...

The Ising model undergoes a phase transition between an ordered and a disordered phase in 2 dimensions or more. In 2 dimensions, the Ising model has a strong/weak duality (between high temperatures and low ones) called the Kramers-Wannier duality. The fixed point of this duality is at the second-order phase transition temperature. In physics, a phase transition is the transformation of a thermodynamic system from one phase to another. ... In quantum field theory and statistical mechanics, a system can be in two possible phases: an ordered phase and a disordered phase. ... In quantum field theory and statistical mechanics, a system can be in two possible phases: an ordered phase and a disordered phase. ... In theoretical physics, S-duality (also a strong-weak duality) is an equivalence of two quantum field theories, string theories, or M-theory. ... Temperature is the physical property of a system which underlies the common notions of hot and cold; the material with the higher temperature is said to be hotter. ... In mathematics, a fixed point of a function is a point that is mapped to itself by the function. ... In physics, a phase transition is the transformation of a thermodynamic system from one phase to another. ...

While the Ising model is an extremely simplified description of ferromagnetism, its importance is underscored by the fact that other systems can be mapped exactly or approximately to the Ising system. The grand canonical ensemble formulation of the lattice gas model, for example, can be mapped exactly to the canonical ensemble formulation of the Ising model. The mapping allows one to exploit simulation and analytical results of the Ising model to answer questions about the related models. In statistical mechanics, the grand canonical ensemble is a statistical ensemble, that means a set of identically prepared systems, each of which is in equilibrium with an external bath with respect to particle and energy exchange. ... A canonical ensemble in statistical mechanics is an ensemble of dynamically similar systems, each of which can share its energy with a large heat reservoir, or heat bath. ...

The Ising model in two dimensions, and in the absence of an external magnetic field, was analytically solved in 1944 by Lars Onsager. Lars Onsager (November 27, 1903 – October 5, 1976) was a Norwegian physical chemist, winner of the 1968 Nobel Prize in Chemistry. ...

See also

• Kuramoto model
• XY model
• Potts model
• Maximal evenness
• Hopfield net

The Kuramoto model, first proposed by Yoshiki Kuramoto (蔵本 由紀 Kuramoto Yoshiki), is a mathematical model for the behavior of a large set of coupled oscillators, and synchronization in general. ... Like the Ising model, the XY model is one of the many highly simplified models in the branch of physics known as statistical mechanics. ... In statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. ... In diatonic set theory maximal evenness is the quality of a collection or scale which for every generic interval there are is either one or two consecutive (adjacent) specific intervals, in other words a scale which is spread out as much as possible. ... A Hopfield net is a form of recurrent neural network invented by John Hopfield. ...

Further reading

• Clough, John; Douthett, Jack; and Krantz, Richard (2000). "Maximally Even Sets: A Discovery in Mathematical Music Theory is Found to Apply in Physics", Bridges: Mathematical Connections in Art, Music, and Science, p.193-200, Conference Proceedings 2000, ed. Reza Sarhangi. Winfield, Kansas: Central Plain Book Manufacturing.

External links

• Barry A. Cipra, "The Ising model is NP-complete", SIAM News, Vol. 33, No. 6; online edition (.pdf)
  • Reports why the Ising model can't be solved exactly in general, since non-planar Ising models are NP-complete.
• Monte Carlo simulations: a java applet implementing various methods.
• Science World article on the Ising Model
• An Ising Applet by Syracuse University
• Phase transitions on lattices
• Nature news article: The Ising on the cake
• Three-dimensional proof for Ising Model impossible, Sandia researcher claims
• Calculation of the Metropolis and the Glauber Transition Probabilities for the Ising Model and for the q-state Potts Model
• Computational Studies of Pure and Dilute Spin Models